
EXAMPLE

Triangle begins:
n\k 1 2 3 4 5 6 7 8
+
1  1
2  1 4
3  2 11 56
4  3 29 370 5752
5  4 94 2666 82310 2519124
6  6 263 19126 1232770 88117873 6126859968
7  12 968 134902 19119198 2835424200 ? ?
8  20 3416 1026667 307914196 109979838540 ? ? ?
A 3 X 3 square can be tiled by three 1 X 2 pieces and three 1 X 1 pieces in the following ways:
++++ ++++ ++++
           
++++ + +++ ++ ++
          
+++ + +++ + +++ +
        
++++ ++++ ++++
.
++++ ++++ ++++
         
++++ +++ + ++++
         
+++ + ++++ ++++
        
++++ ++++ ++++
.
++++ ++++
     
++++ ++++
     
++++ ++++
     
++++ ++++
The first six of these have no symmetries, so they account for 8 tilings each. The last two has a mirror symmetry, so they account for 4 tilings each. In total there are 6*8+2*4 = 56 tilings. This is the maximum for a 3 X 3 square, so T(3,3) = 56.
The following table shows the sets of pieces that give the maximum number of tilings up to (n,k) = (7,5). The solutions are unique except for (n,k) = (2,1) and (n,k) = (6,1).
\ Number of pieces of size
(n,k)\ 1 X 1  1 X 2  1 X 3  1 X 4
++++
(1,1)  1  0  0  0
(2,1)  2  0  0  0
(2,1)  0  1  0  0
(2,2)  2  1  0  0
(3,1)  1  1  0  0
(3,2)  2  2  0  0
(3,3)  3  3  0  0
(4,1)  2  1  0  0
(4,2)  4  2  0  0
(4,3)  3  3  1  0
(4,4)  5  4  1  0
(5,1)  3  1  0  0
(5,2)  4  3  0  0
(5,3)  4  4  1  0
(5,4)  7  5  1  0
(5,5)  7  6  2  0
(6,1)  2  2  0  0
(6,1)  1  1  1  0
(6,2)  4  4  0  0
(6,3)  7  4  1  0
(6,4)  8  5  2  0
(6,5)  10  7  2  0
(6,6)  11  8  3  0
(7,1)  2  1  1  0
(7,2)  5  3  1  0
(7,3)  8  5  1  0
(7,4)  10  6  2  0
(7,5)  11  7  2  1
